For an introduction to artificial neural networks, see Chapter 1 of myfree online book: http://neuralnetworksanddeeplearning.com/chap1.html

A good series of videos on neural networks is by 3Blue1Brown. Starthere: https://www.youtube.com/watch?v=aircAruvnKk

This video just shows the (very simple!) basic idea of the proof. For the full proof of the universal approximation theorem, including caveats that didn't make it into this video, see Chapter 4 of my book:http://neuralnetworksanddeeplearning.com/chap4.html

This video was made as part of a larger project, on media for mathematics: http://cognitivemedium.com/magic_paper/